Problem: $\dfrac{ 2k + 8l }{ 5 } = \dfrac{ 10k + 8m }{ 9 }$ Solve for $k$.
Answer: Multiply both sides by the left denominator. $\dfrac{ 2k + 8l }{ {5} } = \dfrac{ 10k + 8m }{ 9 }$ ${5} \cdot \dfrac{ 2k + 8l }{ {5} } = {5} \cdot \dfrac{ 10k + 8m }{ 9 }$ $2k + 8l = {5} \cdot \dfrac { 10k + 8m }{ 9 }$ Multiply both sides by the right denominator. $2k + 8l = 5 \cdot \dfrac{ 10k + 8m }{ {9} }$ ${9} \cdot \left( 2k + 8l \right) = {9} \cdot 5 \cdot \dfrac{ 10k + 8m }{ {9} }$ ${9} \cdot \left( 2k + 8l \right) = 5 \cdot \left( 10k + 8m \right)$ Distribute both sides ${9} \cdot \left( 2k + 8l \right) = {5} \cdot \left( 10k + 8m \right)$ ${18}k + {72}l = {50}k + {40}m$ Combine $k$ terms on the left. ${18k} + 72l = {50k} + 40m$ $-{32k} + 72l = 40m$ Move the $l$ term to the right. $-32k + {72l} = 40m$ $-32k = 40m - {72l}$ Isolate $k$ by dividing both sides by its coefficient. $-{32}k = 40m - 72l$ $k = \dfrac{ 40m - 72l }{ -{32} }$ All of these terms are divisible by $8$ Divide by the common factor and swap signs so the denominator isn't negative. $k = \dfrac{ -{5}m + {9}l }{ {4} }$